As far as I'm aware, every proof of this fact is essentially the same as Hempel's original proof. I don't know whether it's "simple" enough for you! The key point is that the fundamental group G of a Seifert-fibred piece has the following property.

**Property.** There exists an integer K such that for any positive integer n there is a finite-index normal subgroup G_{n} of G such that any peripheral subgroup P intersects G_{n} in KnP.

It's not too hard to prove. There's a nice account in a paper by Emily Hamilton (which generalizes Hempel's result).

The other important fact is that peripheral subgroups in Seifert-fibred manifold groups are separable (ie closed in the profinite topology, for any non-experts out there).

Using these two pieces of information, you can piece together finite quotients of Seifert-fibred pieces into a virtually free quotient of π_{1} of the graph manifold in which your favourite element doesn't die.

**Note on separability of peripheral subgroups.** Of course, Scott proved that Seifert-fibred manifold groups are LERF. But, by a pretty argument of Long and Niblo, a subgroup is separable if and only if the double along it is residually finite. In particular, you can deduce peripheral separability from the easier fact that Seifert-fibred manifold groups are residually finite.